Lemma 4.3.2

Let and be -algebras, and let be a -homomorphism. Suppose that belongs to the kernel of .

  1. There exist a natural number , a projection , and a unitary in such that and
    In other words, if is in the kernel of the induced K0 map, then its form in the k theory of B is standard with a projection “p”, and the image of p under the unitized map is unitarily equivalent to its own scalar part

  2. If is surjective, then there is a projection , , and . In other words, the unitary is just 1!.